Teaching Physics Innovatively 2015. Puzzling problems on gravity adi 1 Tam´ as Tasn´ 17. August, 2015. 1 BME, Institute of Mathematics
Contents Introduction Instructive problems Enveloping curve of elliptic orbits Conclusion
Contents Introduction Instructive problems Enveloping curve of elliptic orbits Conclusion
Overview Simple laws = ⇒ complicated motions Toolkit: ◮ Newton’s gravitational law ◮ Kepler’s laws ◮ Conservation laws (energy, angular momentum, momentum) ◮ Geometry of conic sections Most of these problems are discussed in the courses preparing for the IPhO.
Overview Simple laws = ⇒ complicated motions Toolkit: ◮ Newton’s gravitational law ◮ Kepler’s laws ◮ Conservation laws (energy, angular momentum, momentum) ◮ Geometry of conic sections Most of these problems are discussed in the courses preparing for the IPhO.
Overview Simple laws = ⇒ complicated motions Toolkit: ◮ Newton’s gravitational law ◮ Kepler’s laws ◮ Conservation laws (energy, angular momentum, momentum) ◮ Geometry of conic sections Most of these problems are discussed in the courses preparing for the IPhO.
Contents Introduction Instructive problems Enveloping curve of elliptic orbits Conclusion
Long pendulum Problem: Find the period T of a mathematical pendulum whose length L is comparable to the radius of the Earth R . (Assumptions: small angular deviation; small distance from the Earth.) Solution: ◮ | T | ≈ | F grav | ≈ mg ���� ���� ◮ T and F grav are not vertical α L ◮ Equation of motion ( α, β ≪ 1): g mL ¨ α = − mgL ( α + β ) , α L = β R m ◮ Result: � � R L R L L →∞ T = 2 π − → 2 π R + L g g Conclusion: Approximate separately the magnitude and the direction of a vector(field).
Long pendulum Problem: Find the period T of a mathematical pendulum whose length L is comparable to the radius of the Earth R . (Assumptions: small angular deviation; small distance from the Earth.) Solution: ◮ | T | ≈ | F grav | ≈ mg ���� ���� ◮ T and F grav are not vertical α L ◮ Equation of motion ( α, β ≪ 1): ~ mg T ~ mL ¨ α = − mgL ( α + β ) , α L = β R m ~ F grav ~ mg ◮ Result: β R � � R L R L L →∞ Earth T = 2 π − → 2 π R + L g g Conclusion: Approximate separately the magnitude and the direction of a vector(field).
Long pendulum Problem: Find the period T of a mathematical pendulum whose length L is comparable to the radius of the Earth R . (Assumptions: small angular deviation; small distance from the Earth.) Solution: ◮ | T | ≈ | F grav | ≈ mg ���� ���� ◮ T and F grav are not vertical α L ◮ Equation of motion ( α, β ≪ 1): ~ mg T ~ mL ¨ α = − mgL ( α + β ) , α L = β R m ~ F grav ~ mg ◮ Result: β R � � R L R L L →∞ Earth T = 2 π − → 2 π R + L g g Conclusion: Approximate separately the magnitude and the direction of a vector(field).
Total energy of elliptic orbits m Problem: An object of mass m is orbiting another object of mass a b a M ≫ m . Express the total me- M c A P chanical energy E ( a , b ) in terms of the major and minor axes a and b . Solution: ◮ Distance of perihelion P and aphelion A : a 2 = b 2 + c 2 r P = a − c , r A = a + c , ◮ Energy conservation: E ( a , b ) = mv 2 r P = mv 2 − G mM − G mM P A 2 2 r A ◮ Angular momentum conservation: mv P r P = mv A r A ◮ Result: E ( a ) = − mMG 2 a Conclusion: The total energy of an elliptic orbit depends only on the major axis a .
Total energy of elliptic orbits m Problem: An object of mass m is orbiting another object of mass a b a M ≫ m . Express the total me- M c A P chanical energy E ( a , b ) in terms r r P A of the major and minor axes a and b . Solution: ◮ Distance of perihelion P and aphelion A : a 2 = b 2 + c 2 r P = a − c , r A = a + c , ◮ Energy conservation: E ( a , b ) = mv 2 r P = mv 2 − G mM − G mM P A 2 2 r A ◮ Angular momentum conservation: mv P r P = mv A r A ◮ Result: E ( a ) = − mMG 2 a Conclusion: The total energy of an elliptic orbit depends only on the major axis a .
Total energy of elliptic orbits m Problem: An object of mass m is orbiting another object of mass a b a M ≫ m . Express the total me- M c A P chanical energy E ( a , b ) in terms r r P A of the major and minor axes a and b . Solution: ◮ Distance of perihelion P and aphelion A : a 2 = b 2 + c 2 r P = a − c , r A = a + c , ◮ Energy conservation: E ( a , b ) = mv 2 r P = mv 2 − G mM − G mM P A 2 2 r A ◮ Angular momentum conservation: mv P r P = mv A r A ◮ Result: E ( a ) = − mMG 2 a Conclusion: The total energy of an elliptic orbit depends only on the major axis a .
Deviation angle of hyperbolic orbits v 0 Problem: A comet passes by the Sun. p Determine its angle of deviation α in terms of the initial speed v 0 and the im- α pact parameter p . Hint: ◮ Apply energy and angular momentum conservation for the perihelion P and the point at infinity ◮ Use the geometry of the hyperbola: � α � = b c 2 = a 2 + b 2 , tan a , p = a , PF = c − b 2
Deviation angle of hyperbolic orbits b a Problem: A comet passes by the Sun. c Determine its angle of deviation α in α/2 terms of the initial speed v 0 and the im- P F α pact parameter p . Hint: ◮ Apply energy and angular momentum conservation for the perihelion P and the point at infinity ◮ Use the geometry of the hyperbola: � α � = b c 2 = a 2 + b 2 , tan a , p = a , PF = c − b 2
Racing satellites Problem: Two satellites, A and B or- bit the Earth on the same circular orbit, A B lags behind A . How should B use B its rocket in order to catch up with A ? Earth (Assume that the rocket can give only a quick impulse to the satellite.) Solution: ◮ If B increases its speed = ⇒ E ( a ) = − mMG < 0 increases = ⇒ 2 a a increases = ⇒ period T increases WRONG!!! ◮ If B decreases its speed = ⇒ E ( a ) decreases = ⇒ a decreases = ⇒ T decreases CORRECT!!!
Racing satellites Problem: Two satellites, A and B or- bit the Earth on the same circular orbit, A B lags behind A . How should B use B its rocket in order to catch up with A ? F E (Assume that the rocket can give only a quick impulse to the satellite.) Solution: ◮ If B increases its speed = ⇒ E ( a ) = − mMG < 0 increases = ⇒ 2 a a increases = ⇒ period T increases WRONG!!! ◮ If B decreases its speed = ⇒ E ( a ) decreases = ⇒ a decreases = ⇒ T decreases CORRECT!!!
Racing satellites Problem: Two satellites, A and B or- bit the Earth on the same circular orbit, A B lags behind A . How should B use B its rocket in order to catch up with A ? E F (Assume that the rocket can give only a quick impulse to the satellite.) Solution: ◮ If B increases its speed = ⇒ E ( a ) = − mMG < 0 increases = ⇒ 2 a a increases = ⇒ period T increases WRONG!!! ◮ If B decreases its speed = ⇒ E ( a ) decreases = ⇒ a decreases = ⇒ T decreases CORRECT!!!
Stopping the Moon Problem: Imagine that the Moon’s orbital motion around the Earth is suddenly stopped. How long would it take for the Moon to fall into the Earth? Remark: The direct integration is beyond the secondary school level. Idea: Apply Kepler’s third law for the two orbits of the Moon.
Stopping the Moon Problem: Imagine that the Moon’s orbital motion around the Earth is suddenly stopped. How long would it take for the Moon to fall into the Earth? Remark: The direct integration is beyond the secondary school level. Idea: Apply Kepler’s third law for the two orbits of the Moon.
Stopping the Moon Problem: Imagine that the Moon’s orbital motion around the Earth is suddenly stopped. How long would it take for the Moon to fall into the Earth? Remark: The direct integration is beyond the secondary school level. Idea: Apply Kepler’s third law for the two orbits of the Moon.
Motions observed from a space station Problem: A space station is orbit- y ing the Earth on a circular trajec- x tory, facing always with the same side towards the Earth. A small ω y object is thrown out of the space space Earth station station with a small initial veloc- ity. How does the object move x R relative to the space station? Remarks: ◮ Solve the problem in rotating reference frame ◮ Approximate the magnitude and the direction of the force vectors separately
Motions observed from a space station Problem: A space station is orbit- y ing the Earth on a circular trajec- x tory, facing always with the same side towards the Earth. A small ω y object is thrown out of the space space Earth station station with a small initial veloc- ity. How does the object move x R relative to the space station? Remarks: ◮ Solve the problem in rotating reference frame ◮ Approximate the magnitude and the direction of the force vectors separately
Contents Introduction Instructive problems Enveloping curve of elliptic orbits Conclusion
The problem Let A be a fixed point in space at a distance d from a fixed sun S of mass M . Particles of mass m are shot from A in different directions at constant speed v . Which points can be reached by the particles? v m M A S d The enveloping curve of the orbits is to be found.
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